Question

Find the mode and median of the following frequency distribution:

$x$	$10$	$11$	$12$	$13$	$14$	$15$
$f$	$1$	$4$	$7$	$5$	$9$	$8$

Answer 1

$x$	$10$	$11$	$12$	$13$	$14$	$15$
$f$	$1$	$4$	$7$	$5$	$9$	$8$

for a frequency distribution,

So, total number of observation $=(1+4+7+5+9+8)=34$

It is a even number.

So, median is the value of average of values at ${\left(\frac{n}{2}\right)}^{th}$ and ${\left(\frac{n+2}{2}\right)}^{th}$.

$\therefore$ Here, median is the average of the values at ${\left(\frac{34}{2}\right)}^{th}$ or ${17}^{th}$ position and ${\left(\frac{34+2}{2}\right)}^{th}$ or ${18}^{th}$ position.

$x$	$10$	$11$	$12$	$13$	$14$	$15$
$f$	$1$	$4$	$7$	$5$	$9$	$8$
$position$	$1$	$1+4=5$	$5+7=12$	$12+5=17$	$17+9=26$	$26+8=34$

So, value of $x$ at ${17}^{th}$ position is $13$ and at ${18}^{th}$ position is $14$.

$\therefore Median=\left(\frac{13+14}{2}\right)=\frac{27}{2}=13.5$.

Mode for a frequency distribution is the value which is the most common frequent value.

Here $14$ has frequency $9$ which is highest. So, mode for the frequency distribution is $14$.